Closed Unit Ball Question?

[Kelly]

So my friends and I have been trying to figure this out forever but we can't seem to get it at all, any help?

Define the closed unit ball [FONT=MathJax_Math-italic]B[FONT=MathJax_Main]⊂ [/FONT][FONT=MathJax_AMS]Rⁿ[/FONT] by:[/FONT]
B:={x∈R∣∥x∥≤1}, where ∥⋅∥ denotes the Euclidean norm on [FONT=MathJax_AMS]Rⁿ, i.e. [FONT=MathJax_Main]∥[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]∥[/FONT][FONT=MathJax_Main]= sqrt (<x,x>) for x[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_AMS]Rⁿ
[/FONT][/FONT]
Prove that:
[FONT=MathJax_Math-italic]P_b =
[/FONT]

• [FONT=MathJax_Math-italic](x/||x||) if ||x|| > 1 [/FONT]
• [FONT=MathJax_Math-italic]​(x) if ||x|| less than or equal to 1.[/FONT]

 

[stapel]

So my friends and I have been trying to figure this out forever but we can't seem to get it at all, any help?

Define the closed unit ball [FONT=MathJax_Math-italic]B[FONT=MathJax_Main]⊂ [/FONT][FONT=MathJax_AMS]Rⁿ[/FONT] by:[/FONT]
B:={x∈R∣∥x∥≤1}, where ∥⋅∥ denotes the Euclidean norm on [FONT=MathJax_AMS]Rⁿ, i.e. [FONT=MathJax_Main]∥[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]∥[/FONT][FONT=MathJax_Main]= sqrt (<x,x>) for x[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_AMS]Rⁿ
[/FONT][/FONT]
Prove that:
[FONT=MathJax_Math-italic]P_b =
[/FONT]

• [FONT=MathJax_Math-italic](x/||x||) if ||x|| > 1 [/FONT]
• [FONT=MathJax_Math-italic]​(x) if ||x|| less than or equal to 1.[/FONT]

Since you and your friends "have been trying to figure this out forever", you'll have at least a few attempts which you can provide. Kindly please reply with this information, so the volunteers can see what you're thinking (you may be right!) and how far you've gotten (you may be almost done!). Thank you!

 

[pka]

Define the closed unit ball [FONT=MathJax_Math-italic]B[FONT=MathJax_Main]⊂ [/FONT][FONT=MathJax_AMS]Rⁿ[/FONT] by:[/FONT]
B:={x∈R∣∥x∥≤1}, where ∥⋅∥ denotes the Euclidean norm on [FONT=MathJax_AMS]Rⁿ, i.e. [FONT=MathJax_Main]∥[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]∥[/FONT][FONT=MathJax_Main]= sqrt (<x,x>) for x[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_AMS]Rⁿ[/FONT][/FONT]
Prove that:
[FONT=MathJax_Math-italic]P_b =
[/FONT]

• [FONT=MathJax_Math-italic](x/||x||) if ||x|| > 1 [/FONT]
• [FONT=MathJax_Math-italic]​(x) if ||x|| less than or equal to 1.[/FONT]

We cannot help unless you tell what \(\displaystyle P_b\) means??